HOMEWORK 5 FOR 18.100B AND 18.100C, FALL 2010
Paper solutions Due 11AM, Thursday, October 21, in lecture or 2-108, Electronic
submission to rbm at math dot mit dot edu by 5PM.
HW5.1 Modified version of Rudin Ch 3 No 1. Prove that for a sequence sn in C, the
convergence of sn implies the convergence of |sn |. Give a counterexample
to the converse statement.
∞ √
P
P
an
HW5.2 Rudin Ch 3 No 7. Show that if an ≥ 0 and
an converges then
n
n
n=1
converges.
√
HW5.3 Rudin Ch 3 No 16. Fix a positive number α. Choose x1 > α, and define
a sequence x2 , x3 ,. . . by the recursion formula
1
α
xn+1 = (xn +
).
2
xn
√
(a) Prove that xn √
decreases monotonically and that limn→∞ xn = α.
(b) Set n = xn − α and and show that
n+1 =
√
so that if β = 2 α then
2n
2
< √n
2xn
2 α
1 2 n
) .
β
(c) This is a good algorithm for computing square roots, since the recursion formula is simple and the convergence is extremely rapid. For
example, if α = 3 and x1 = 2, show that 1 /β < 1/10 and therefore
n+1 < β(
5 < 4 · 10−16 , 6 < 4 · 10−32 .
HW5.4 Let f : X −→ Y be a map between sets. Let P(X) and P(Y ) be the power
sets – the collection of subsets respectively of X and Y. Define maps
f# : P(X) −→ P(Y ), f# (A) = {y ∈ Y ; ∃ a ∈ A with y = f (a)}
f # : P(Y ) −→ P(X), f # (B) = {x ∈ X; f (x) ∈ B}
(usually denoted as f and f −1 respectively). Compute the two composite
maps f # ◦ f# : P(X) −→ P(X) and f# ◦ f # : P(Y ) −→ P(Y ).
HW5.5 Show that for each fixed point p in a metric space X the distance from p,
f (x) = d(x, p) defines a continuous function f : X −→ R.
1